Carrier aggregation

ABSTRACT

A method implemented in a mobile communications system is disclosed. The method includes selecting primary component carrier c p  for a new user according to a formula, and performing joint secondary CC selection for one or more Long Term Evolution (LTE)-Advanced users and resource scheduling for one or more LTE users and said one or more LTE-Advanced users. Other methods, systems, and apparatuses also are disclosed.

This application claims the benefit of U.S. Provisional Application No.61/675,511, entitled “Energy Efficient Carrier Aggregation for LTE-AdvSystems,” filed on Jul. 25, 2012, the contents of which are incorporatedherein by reference.

BACKGROUND OF THE INVENTION

The present invention relates to carrier aggregation and, moreparticularly, to secondary component carrier allocation.

Carrier aggregation (CA) is an important feature of LTE-advanced thatallows its users to aggregate up to 100 MHz of (dis-)contiguous spectralchunks to provide increased data rates. While the conventional approachof allowing LTE-adv users to be configured on all component carriers,results in maximum diversity gain for scheduling, it also increases theusers' power consumption and processing that scale with the number ofcomponent carriers. We argue that it is possible to operate the LTE-advusers on a small subset of component carriers to reduce their energyconsumption, without any appreciable loss to the scheduling gain. A stepin realizing this goal however, is to address the joint problem ofcomponent carrier selection as well as scheduling.

We highlight the hardness of the joint problem when the number ofcomponent carriers that can be activated for a LTE-adv user is limited.Towards solving the problem, we consider various models that incorporatecontiguous/dis-contiguous CA as well as backlogged/finite buffers andpropose efficient, greedy algorithms with performance guarantees thatare also simple to implement. Our evaluations based on LTE simulationparameters, reveal that our algorithms help realize 80-90% of themaximum scheduling gain with just half the component carriers andprovide 15-25% throughput gain over conventional load and signal power(RSRP) based carrier selection schemes.

The conventional approach of allowing LTE-adv users to be configured onall component carriers, results in maximum diversity gain forscheduling. However, it also increases the users' power consumption andprocessing that scale with the number of component carriers.

The proliferation of mobile devices and the exponential growth of mobiledata traffic has increased the demand for higher data rates from nextgeneration cellular networks like LTE-advanced, WiMAX, etc. In additionto OFDMA being employed as the air interface in all these technologies,several other features such as small cells, carrier aggregation, etc.are being considered. While small cells increase the area spectralefficiency and are a key to increasing the system capacity, severalchallenges remain in realizing them in practice. On the other hand,carrier aggregation provides an immediate, effective solution fornetwork operators to repurpose spectrum from older technologies (eg.2/3G to 4G) and aggregate fragmented spectral allocations to deliverhigher data rates.

Carrier aggregation (CA) can be of multiple types as shown in FIG. 1(a).Component carriers (CC, spectral chunks) can be aggregateddis-contiguously either within a band (intra-band) or across bands(inter-band), but contiguously only within a band (intra-band). While CAis supported only by LTE adv users, LTE-adv (release 10 onwards) itselfallows for backward compatibility with release 8/9 users that operate ononly one CC. For every user, a CC is configured to be the primary CC(PCC) that is responsible for key operations such as locationregistration, RRC (re-)establishment, etc. and hence cannot be changeddynamically. On the other hand, the additional CCs (secondary CCs) in CAcan be (de-)activated dynamically for LTE-adv users.

In the conventional approach, where LTE-adv users are configured withall available CCs, the selection of CCs is restricted to the choice ofPCC for each user, with the remaining CCs serving as SCCs. Due to thenature of operations on PCC, its selection is decoupled from schedulingand determined semi-statically based on load-balancing or referencesignal received power (RSRP). While activating LTE-adv users on all CCsprovides maximum diversity gain through scheduling, it also increasesthe energy consumption and processing at the user (device)—factors thatscale with the number of CCs activated. Hence, we posit the followingquestion: Is it possible to operate the LTE-adv users on a small subsetof component carriers to reduce their energy consumption, without anyappreciable loss to the scheduling gain? Given the plethora of networkinterfaces and applications being housed by smart mobile devices andtheir consequent impact on battery drainage, understanding the answer tothe above question is both important and timely.

We answer in the affirmative and argue that it is indeed possible tooperate the LTE-adv users on a small subset of CCs without anappreciable loss to scheduling performance. Note that selection ofsecondary CCs (SCCs) for LTE adv users now becomes an integral componentand directly impacts scheduling performance. Given that SCCs can be(de-)activated dynamically, a key to keeping the loss in performancesmall, is to integrate and couple CC selection with scheduling andaddress them jointly for LTE-adv users. Towards addressing this goal andhence seeking an answer to our motivating question, we make thefollowing contributions:

-   -   We prove the hardness of the coupled problem of CC selection and        scheduling when the number of CCs that can be activated for a        LTE-adv user is limited.    -   Towards solving the problem, we consider various models that        incorporate contiguous(C)/dis-contiguous(D) CA as well as        backlogged(B)/finite(F) user buffers and propose efficient,        greedy algorithms with performance guarantees that are also        simple to implement. Specifically, our algorithms yield        approximation guarantees of ½, ¼, ½, and ⅓ for the models DB,        DF, CB and CF respectively.    -   Our evaluations based on LTE simulation parameters, reveal that        our algorithms help realize 80-90% of the maximum scheduling        gain with just half the component carriers and provide 15-25%        throughput gain over conventional load-based and RSRP-based        carrier selection schemes.

Our results are promising and indicate that with the help of efficientlydesigned joint CC selection and scheduling algorithms for LTE-adv users,it is possible to realize close-to the full performance benefits of CA(achieved with all CCs), while expending only a fraction of the userenergy.

Existing solutions [1, 2, 3] restrict the number of component carriersfor a user by load balancing users on different component carrriers(CCs). However, since the allocation of specific CCs to users is doneindependent of scheduling, it comes at the expense of diversity gain andthroughput performance.

REFERENCES

-   [1] R. Ratasuk, D. Tolli, and A. Ghosh, “Carrier aggregation in lte    advanced,” in IEEE VTC, May 2010.-   [2] L. Garcia, K. Pedersen, and P. Mogensen, “Autonomous component    carrier selection: Interference management in local area    environments for lte-advanced,” in IEEE Communications Magazine,    September 2009.-   [3] A. Li, K. Takeda, N. Miki, Y. Yan, and H. Kayama, “Search space    design for cross-carrier scheduling in carrier aggregation of    lte-advanced system,” in IEEE ICC, June 2011.

BRIEF SUMMARY OF THE INVENTION

An objective of the present invention is to aim to jointly performed CCselection for users along with their resource scheduling. This helpsdetermine the limited but appropriate set of CCs for each user. Whileenergy consumption is lowered due to the reduced use of CCs, diversitygain and hence throughput performance is also not very sacrificed in theprocess.

An aspect of the present invention includes a method implemented in amobile communications system. The method comprises selecting primarycomponent carrier c_(p) for a new user according to formula

${c_{p} = {\arg\;{\max_{c}\frac{r_{{avg},c}}{l_{c}}}}},$where (r_(k,c) ^(avg), is reference signal received power (RSRP)averaged over a spectrum for user k on component carrier c, andperforming joint secondary CC selection for one or more Long TermEvolution (LTE)-Advanced users and resource scheduling for one or moreLTE users and said one or more LTE-Advanced users.

Another aspect of the present invention includes a mobile communicationssystem. The mobile communications system comprises a first selectionunit to select primary component carrier c_(p) for a new user accordingto formula

${c_{p} = {\arg\;{\max_{c}\frac{r_{{avg},c}}{l_{c}}}}},$where (r_(k,c) ^(avg), is reference signal received power (RSRP)averaged over a spectrum for user k on component carrier c, and aperformance unit to perform joint secondary CC selection for one or moreLong Term Evolution (LTE)-Advanced users and resource scheduling for oneor more LTE users and said one or more LTE-Advanced users.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 depicts (a) types of CA and (b) dis-contiguous CA with 10 LTE-advusers.

FIG. 2 depicts performance of CA algorithms, especially with regard to(a) an impact of the number of CCs, (b) gain over load-based, (c) gainover RSRP-based, and (d) an impact of the number of LTE-Adv users.

FIG. 3 depicts an impact of Rel 8/9 users and finite buffers, especiallywith regard to (a) an impact of Rel 8/9 users, (b) gain over load-based,(c) gain over load-based (finite), and (d) gain over RSRP-based(finite).

FIG. 4 depicts an approach proposed in this document.

FIG. 5 depicts an approach for the joint CC selection in case ofbacklogged user buffers.

FIG. 6 depicts an approach for the joint CC selection in case of finiteuser buffers.

DETAILED DESCRIPTION

We provide a light-weight (low complexity) solution for joint CCselection and scheduling for LTE-advanced users. This allows for bothbetter throughput performance of LTE-advanced users while incurring onlya fraction of the energy compared to conventional schemes. Further, thelow complexity nature of the solutions allow for easier realization atthe base stations.

II. System Description

A. Carrier Aggregation

To provide high data rates to next generation mobile devices, it becomesin-dispensable for network operators to aggregate several fragmentedspectral chunks—a process referred to as carrier aggregation (CA). WhileCA and hence our contributions are common to next generationtechnologies (LTE-A, WiMAX) in general, we will describe them in thecontext of the more popular one, namely LTE-A.

CA can be of multiple types as shown in FIG. 1(a). Contiguous ordis-contiguous component carriers (CC) within a band (intra-band) can beaggregated, while only dis-contiguous CCs can be aggregated across bands(inter-band) by definition (for eg. CCs in 700 MHz and 2 GHz). LTE-A(3GPP release 10 onwards) allows for up to 5 CCs of 20 MHz maximumbandwidth to be aggregated, thereby allowing for LTE-A users to operatewith 100 MHz spectrum. LTE-A provides backward compatibility with LTE(3GPP release 8/9) users that operate on only one CC. For every user,when it first (re-)establishes a radio control connection, a CCconfigured to be its primary CC (PCC). The PCC can then be used toconfigure additional CCs called secondary CCs (SCCs) that are accessibleto LTE-A users. Unlike the PCC, the SCCs can be (de-)activateddynamically for LTE-A users during carrier aggregation using a bitmap.

Note that, the power consumption per user increases with the number ofCCs (bandwidth size) it has to receive and process for control and datainformation (not just link measurements). Thus, while configuring aLTE-A user with all CCs will provide maximum scheduling diversity gainby allowing the user to be scheduled on any of the resource blocks inall CCs, it will also incur the maximum power consumption. Similarly,while contiguous CA might offer lower scheduling diversity compared todis-contiguous CA, it will also potentially incur lower powerconsumption since its implementation can be realized with a single FFTmodule and a RF component unlike the latter. Hence, the RRM (radioresource management) process in LTE-A incorporates a new feature calledCC configuration, whereby each LTE-A user can be configured to operateon a specific set of CCs.

B. Motivation

Given the multitude of interfaces and applications running on smartmobile devices and their power-hungry nature, configuring the LTE-Ausers (using CC configuration feature) to operate on a limited set ofSCells to save power appears to be a natural solution. However, suchenergy savings will have to come at the cost of scheduling performance.Hence, a key question we try to address in this work is whether it ispossible to save on LTE-A users' power consumption without anappreciable loss in scheduling performance?

Current approaches to CA configure a PCC based on load and usercapabilities (decoupled from scheduling), while allowing all remainingCCs to be used as SCCs, purely from the perspective of maximizingscheduling diversity gain. Hence, to understand the tradeoff betweenscheduling diversity gain and power consumption reduction, we study thefraction of the maximum (aggregate) scheduling throughput that can beachieved when the number of CCs configured for LTE-A users is limited(compared to five CCs allowed in release 10). For this purpose, weconsider two baselines, where we apply the PCC selection criteria to SCCselection as well (i.e. based on user load on CCs or average referencesignal received power (RSRP) of CCs), thereby decoupling the latter fromscheduling. We compare them against an integrated scheme, whereselection of SCCs is determined jointly with scheduling. The performanceof these schemes (algorithms employed explained in subsequent sections)is compared against an optimal scheme that allows for access to all CCsfor LTE-A users. The result in FIG. 1(b) clearly illustrates two points:(i) 80-90% of the maximum scheduling gain can be achieved with just halfthe CCs, namely 2-3 CCs, and (ii) to keep the loss in schedulingperformance minimal while saving on user power, one needs to carefullypick the SCCs by integrating the SCC selection process with scheduling.This in turn motivates us to address the joint problem of SCC selectionand scheduling for a limited number of CCs in this work.

C. Related Work

Carrier aggregation, being one of the recent advancements incorporatedin LTE-A, is slowly garnering attention from the research community,which provides an overview of the various options and featuresassociated with CA in LTE-A and how it would coexist with legacy LTEusers. RRM (scheduling) in OFDMA networks with carrier aggregation hasbeen studied recently in the past couple of years. In a system-levelevaluation of CA, CC selection is restricted to only PCC based onload-balancing and LTE-A users are allowed to use all CCs. CA isemployed to alleviate interference in heterogeneous networks (macrocellsand femtocells), where CCs can be selected autonomously by differentnetworks so as to alleviate interference. There is a problem of searchspace design (mapping) for control information signaling in CA withcross-CC scheduling. Another step is to extend MIMO scheduling in LTE towork with CA. To the best of our knowledge, existing works thus far havenot looked at the problem of configuring LTE-A users on a subset of CCsfor energy efficiency purposes, wherein the joint problem of CCselection and scheduling becomes important and challenging.

III. Scheduling with Carrier Aggregation

Recall that there are two aspects to CC selection: PCC (semi-static)selection that is common to both LTE (Rel 8/9) and LTE-A users and SCC(dynamic) selection that is specific to LTE-A users. While our focus ison the joint (dynamic) SCC selection (for LTE-A users) and scheduling(all users), we do need to specify how the PCCs are chosen first as thisimpacts the remaining set of SCCs available for each user.

A. PCC Selection

The choice of PCC is made either based on user load or received signalpower (RSRP) on the CCs in existing schemes.

-   -   Load-based: The CC that has the smallest number of users (l_(c))        configured on it thus far is chosen as the PCC for a given user        k (P_(k)=arg min_(c)l_(c)).    -   RSRP-based: The CC that yields the highest average RSRP (r_(k,c)        ^(avg), averaged over the spectrum) for a given user (k) is        chosen as its PCC (P_(k)=arg max_(c)r_(k,c) ^(avg)).

The load-based scheme allows for load balancing and utilization of allCCs but is agnostic to the rates seen by the users on the CCs, while theroles are reversed for the RSRP-based scheme. Hence, to strike a balancebetween the two factors, we employ PCC selection based on the followingmetric:

$c_{p} = {\arg\;{\max_{c}{\frac{r_{{avg},c}}{l_{c}}.}}}$B. Joint SCC Selection and Scheduling

1) Scheduling Model: We consider a downlink, OFDMA system as in LTE,where data transmissions occur in frames. Every downlink frame is atwo-dimensional structure of symbols and sub-channels. Resourceallocations to users are made in the granularity of resource blocks(RBs), where an RB spans multiple sub-channels and all symbols in theframe.

The objective of our scheduling algorithms is to maximize the end-to-endsystem throughput subject to the popular proportional fairness (PR)model (maxΣ_(k)β_(k) log r _(k)), where β_(k) captures the priorityweight of user's QoS class and r _(k) its average throughput. The systemsolution can be shown to converge to the optimum PF allocation at longertime scales if the base station (BS) scheduler's decisions at each frameare made to maximize the aggregate marginal utility, S_(max)=argmax_(s){Σ_(kεS)ΔU_(k)}. ΔU_(k) denotes the marginal utility received byuser k in a feasible schedule S and is given by

$\frac{\beta_{k}r_{k}^{CA}}{{\overset{\_}{r}}_{k}}$for PF, where r_(k) ^(CA) is the instantaneous rate received by the userin the frame in the presence of carrier aggregation.

Thus, at each frame t, user weight

${v_{k}(t)} = \frac{\beta_{k}}{{\overset{\_}{r}}_{k}(t)}$varies with r _(k)(t) and accounts for both fairness and QoS. Thescheduling problem at the BS then reduces to determining the frameschedule that maximizes the following aggregate weighted rate subject todesired CA and resource allocation constraints.

$\begin{matrix}{{S_{\max}(t)} = {\arg\;{\max\limits_{S}{\sum\limits_{k \in S}{{v_{k}(t)} \cdot {r_{k}^{CA}(t)}}}}}} & (1)\end{matrix}$

2) Problem Formulation: The rate received by a user in a_(k) frame(r^(CA)) depends on the set of RBs allocated to it and the rate obtainedon those RBs. This in turn depends on the type of the user (LTE orLTE-A) as well as the CCs assigned to it. While the PCCs arepre-determined for all users (Sec. III-A), we still need to address thejoint problem of SCC selection (for LTE-A users with limited number ofCCs) and scheduling (for all users), which can be formally stated asfollows.

${SCA}\text{:}\mspace{14mu}{Maximize}{\sum\limits_{k}{v_{k}{\sum\limits_{c \in {??}}{\sum\limits_{m \in \mathcal{M}}{y_{k,m,c}x_{k,c}r_{k,m,c}}}}}}$${{s.t.\mspace{14mu}{\sum\limits_{c}x_{k,c}}} \leq \left\{ {n,1} \right\}},{\forall{k \in \left\{ {{{LTE}\text{-}A},{LTE}} \right\}}}$${{\sum\limits_{k}{y_{k,m,c}x_{k,c}}} \leq 1},{\forall m},c$${{\sum\limits_{c,m}{y_{k,m,c}x_{k,c}r_{k,m,c}}} \leq B_{k}},{\forall k}$x_(k, c) = 1, if  x_(k, c − 1) ⋅ x_(k, c + 1) = 1, ∀c, k ∈ LTE-Ax_(k,c) and y_(k,m,c) are indicator variables indicating the allocationof CC c to user k and the allocation of RB m in CC c to user k (whosecorresponding rate is r_(k,m,c))respectively.

and

represent the set of CCs and RBs in each CC, with |

|=N and |

|=M.

The first constraint limits the total number of CCs that can beallocated to LTE-A and LTE users to n and 1 respectively. Note that oneof the CCs will be pre-assigned (PCC selection, x_(k,P) _(k) =1) for allusers. Hence, SCCs need to be selected only for LTE-A users. The secondconstraint captures the conflict-free assignment of RBs in each CC tousers. The third constraint is specific to the case, where users havefinite data buffers (B_(k), corresponding to short-lived data sessions)that limits their net rate allocation. The final constraint is specificto the case, where LTE-A users must be allocated contiguous CCs (forcontiguous CA). While this constraint reduces the scheduling flexibilityin contiguous CA compared to dis-contiguous CA, it allows for lowerpower consumption in the former since its implementation can be realizedwith a single FFT module and a RF component unlike the latter. Thus,depending on the nature of CA (contiguous, C or dis-contiguous, D) anduser buffers (backlogged, B or finite, F), we can have four models underwhich the problem can be addressed, namely DB, FB, CB and CF.

While the current fairness model between LTE-A and LTE users is based onnet throughput (over al CCs), we discuss how our proposed algorithms andguarantees would also apply to an alternate model favoring LTE-A usersby considering per-CC fairness.

C. Problem Hardness

Theorem 3.1: SCA1 is NP-hard to solve.

We consider a simpler instance of the problem (SCA), namely withbacklogged buffers and CC limit n=1, which is a special case of all thefour models BD, BC, FD, FC. We prove its hardness by giving apolynomial-time reduction from the edge-2-colorable problem, therebyautomatically establishing the hardness of the four models.

Given the hardness of our problem, we will now focus on designingefficient algorithms with approxi-mation guarantees that are alsoeasy-to-implement for each of the four models in subsequent sections.

IV. Dis-Contiguous Carrier Aggregation

In discontiguous CA, users are not constrained in picking CCs that arecontiguous.

A. Backlogged User Buffers

Given a PCC assignment for each user (Sec. III-A), we need to selectSCCs for LTE-A users as well as assign RBs in CCs to all the users(subject to their CC assignment) so as to maximize our objective ofweighted sum rate. Considering users with backlogged user buffers makesit easier to handle the problem, where we can focus on users thatprovide the best rate on each RB without any buffer under-flow concerns.However, the hardness still remains, due to the limit on the number ofCCs that can be assigned to users. We propose the following greedyalgorithm (GCA-BD) to address the problem.

Algorithm 1 Greedy Scheduler for SCA-BD: GCA-BD  1: Input: CC limit n;PCC List 

 = {(k, P_(k))}, ∀k ; rate r_(k,m,c), ∀k, m, c  2:

 ←

_(lteA) ∪

_(lte);

_(lteA) ← {1, . . . ,|

_(lteA) |},

_(lte) ← {|

_(lteA) | +1, . . . ,|

 |} ;

 ← Ø  3: Define

_(lteA)′ ← {k_(l) : k ∈ [1,|

_(lteA) |], l ∈ [1, n − 1]}, k_(l) = k, ∀l  4: Define φ_(u) ← {(u,c) : c∈ [1, N]} \

, ∀u ∈

_(lteA)′  5: while

_(lteA)′ ≠ Ø do  6:  f(

) = Σ_(c)Σ_(m=1) ^(M)max_(u:(u,c)∈ )

_(∪)

{v_(u)r_(u,m,c)}  7:  f(

∪(u′, c′)) = Σ_(c,m)max_(u:(u,c)∈)

_(∪)

_(∪(u′,c′)){v_(u)r_(u,m,c)}  8:  (u*, c*) = arg max_((u′,c′)∈φ) _(u′)_(:u′∈ )

{f(

∪(u′,c′)) − f(

)}  9:  

 ←

∪(u*, c*) ;

_(lteA)′ ←

_(lteA)′\u* ;  φ_(u) ← φ_(u) \ (u*, c*), ∀u = u* 10: end while 11:k_(m,c)* = arg max_(k:(k,c)∈)

_(∪ )

{v_(k)r_(k,m,c)}, ∀m, c

From the given set of LTE-A users (

_(lteA)) a virtual user set (

′_(lteA)) is formed, where each LTE-A user is replicated n−1 times,where n is the limit on the number of CCs (including PCC) that can beassigned to it (steps 2,3). Now the problem reduces to selecting one CC(other than PCC, from φ_(u)) for each user (u) in (

′_(lteA)) step 4). This in turn is determined greedily by finding theuser-CC pair that yields the highest marginal utility (steps 6-8). Sinceusers have backlogged buffers, the utility of an assignment amounts tofinding the best weighted rate on each RB in a CC based on the users whoare assigned to that CC and aggregating them (steps 6,7). Also notethat, while we are determining CC assignments to LTE-A users, theutilities are determined accounting for both SCC and PCC assignments aswell as LTE-A and LTE users (

∪

). Once a user-CC pair is selected, the remaining set of LTE-A usersthat require SCC assignment is updated (step 9) and the procedurerepeats till all SCC assignments to LTE-A users are made (steps 5-10).Then, based on the final assignment of PCC and SCCs, the allocation ofRBs to users in each CC can be easily computed (step 11). The bulk ofthe time complexity comes from step 8, which runs in O(KNMn). This alongwith the while loop that runs |

_(lteA)|n times, results in a net time complexity of O(K²n²NM).

We will now establish an approximation guarantee for Algorithm GCA-BD.Since most of our algorithms leverage sub-modular maximization toprovide performance guarantees, we first present some relevantdefinitions in this regard.

Partition Matroid: Consider a ground set Ψ and let S be a set of subsetsof Ψ. S is a matroid if, (i) ØεS, (ii) If AεS and B⊂A , then BεS , and(iii) If A,BεS and |A|>|B|, there exists an element xεA\B, such thatB∪{x}εS. A partition matroid is a special case of a matroid, whereinthere exists a partition of Ψ into components, φ₁,φ₂, . . . such thatAεS if and only if |A∩φ_(i)|≦1, ∀i.

Sub-modular function: A function f(·) on S is said to be sub-modular andnon-decreasing if ∀x,A,B such that A∪{x}εS and B⊂A then,f(A∪{x})−f(A)≦f(B∪{x})−f(B)f(A∪{x})−f(A)≧0, and f(∅)=0Theorem 4.1 GCA-BD's worst case performance is within ½ of the optimum.

Proof: The sub-optimality of maximizing a non-decreasing, sub-modularfunction over a partition matroid using a greedy algorithm of the formx=arg max_(xεφ) _(i) f(A∪{x})−f(A) every iteration was shown to bebounded by ½ in [13]. We will now show that GCA-BD is such an algorithm,with our scheduling objective corresponding to a non-decreasing,sub-modular function to obtain the desired result.

Let the ground set be composed of following tuples.Ψ={(u,c):∀uε

′ _(lteA) ,cε[1,N]}\

  (2)

Now Ψ can be partitioned into φ_(u)={(u,c):∀cε[1,N]}\

. Let S be defined on Ψ as a set of subsets of Ψ such that for allsubsets AεS, we have (i) if B⊂A, then BεS; (ii) if element xεA\B, thenQ∪{x}εS; and (iii) |A∩φ_(u)|≦1, ∀u. This means that S is a partitionmatroid. Since the limit of SCCs on each LTE-A user has been translatedto that of one CC for every virtual user in

′_(lteA′); the above conditions enable any AεS to provide a feasibleschedule. This allows S (a partition matroid) to capture all feasibleschedules and hence our scheduling problem. Our scheduling objective isgiven as,

${{f(A)} = {\sum\limits_{c \in {??}}{\mu_{c}(A)}}};$${where},{{\mu_{c}(A)} = {\sum\limits_{m = 1}^{M}{\max\limits_{u:{{({u,c})} \in {A\bigcup{??}}}}\left\{ {v_{u}r_{u,m,c}} \right\}}}}$

It can be seen that if B⊂A, then μ_(i)(B)≦μ_(i)(A). Further, thedifference between A and B is that some CCs have more users assigned tothem in A than in B. Hence, when a new user u is added on CC c, themarginal gain the user can contribute to c is potentially less in A thanin B. Hence, for an element (u,c) such that A∪{(u,c)} forms a validschedule, it follows that f(A∪{(u,c)}−f(A)≦f(B∪{(u,c)})−f(B). Note thatalthough the scheduler focuses only on SCC selection for LTE-A users,f(A) incorporates the utility of both SCC assignments to LTE-A users aswell as PCC assignment to all users. Since the utility of the schedulecorresponding only to the PCC allocation of all users is fixed(constant) and does not impact the SCC selection of LTE-A users (due tobacklogged buffers), removing it from f(A) would allow for normalization(f(Ø)=0). This establishes that the function f(A) is indeed sub-modularand non-decreasing. Further, our scheduling problem aims to maximizethis non-decreasing, sub-modular function over a partition matroid.Thus, by picking the user-CC pair yielding the highest marginal utilityin f(A) in every iteration (steps 6-8), GCA-BD incurs a sub-optimalityof ½ that follows from the result.

B. Finite User Buffers

A key difference with respect to backlogged buffers is that the rate ona RB in a CC for a user is dependent on its own as well as other users'prior allocations as they affect the remaining data in its buffer. Wepropose an algorithm (Algorithm GCA-FD) that considers all users (notjust LTE-A) and assigns SCCs to LTE-A users sequentially in the sensethat all SCC assignments to a user are completed before moving toanother user. As we will show later, such an approach is crucial inestablishing a performance guarantee for the algorithm.

Algorithm 2 Greedy Scheduler for SCA-FD: GCA-FD 1. Input: CC limit n;PCC List

 = {(k, P_(k))}, ∀k ; rate r_(k,m,c), ∀k, m, c; Buffer limit B_(k) , ∀k2.

 ←

_(lteA) ∪

_(lte);

_(lteA) ← {1, . . . ,|

_(lteA)|},

_(lte) ← {|

_(lteA)| + 1, . . . ,|

|};

 ← Ø 3. Define φ_(u) ← {(u,{right arrow over (c)}_(u)):∀u ∈

}; {right arrow over (c)}_(u) ^(′) = <c_(u,1),...,c_(u,n−1)>, c_(u,j) ε[1,N]\P_(u),∀u ε

_(lteA); {right arrow over (c)}_(u) = P_(u), ∀u ε

_(lte) 4. for k ε [1, |

|] do 5.  for u ∉ S & u ε K_(lteA) do 6.   for j ε [1, n − 1] do 7.   c_(u,j) = arg max_(c∉)<c _(u) _(,1),...,c _(u,j) _(−1)∪P) _(u) {f(

 ∪ (u,<c_(u,1),...,c_(u,j−1),c>)) −    f(S ∪(u,<c_(u,l),...,c_(u,j−1)>))} 8.   end for 9.  end for 10.  (u,{rightarrow over (c)}_(u))* = arg max_(u∉)

{f(

 ∪ (u, {right arrow over (c)}_(u))) − f(

)} 11.  

 ←

 ∪ (u, {right arrow over (c)}_(u))* 12. end for 13. Compute f(

) to obtain RB allocation in each CC. 14. 15. Computing f(

′) 16. Let D_(k) = B_(k) , ∀k ∈

;

 = Ø ; U = 0 17. For i ∈ [1, NM] do 18. (u*, m*, c*) = argmax_(u,m,c:(m,c)∉)

_(;(u,c)εS′∪)

{v_(u) min{r_(u,m,c), D_(u)} } 19. U = U + min{r_(u*,m*c*), D_(u*)};D_(u*) = D_(u*) − min{r_(u*,m*,c*),D_(u*)};

 ←

 ∪ (m*, c*) 20. end for 21. Return U

The algorithm comprises of three and two levels of decision making forLTE-A and LTE users respectively, which can be explained in a top-downapproach for easier exposition. At the highest level, it picks the userwith its CC assignment ({right arrow over (c_(u))}: n−1 SCCs for LTE-Auser, 1 fixed PCC P_(u) for all users; step 3) that yield the highhestmarginal utility in each iteration (steps 10-11). However, to computethis, we need to first determine the assignment of n−1 SCCs if the useris an LTE-A user, of which there are exponential (in n) possibilities.Hence, the second level of decision making (only for LTE-A users) is todetermine the assignment of SCCs for each un-assigned LTE-A user, whichin turn is accomplished iteratively for a fixed user by picking the SCCthat yields the highest marginal utility (steps 6-8). However, notethat, utility is always computed with respect to the entire currentallocation (to all users) and not just specific to the given user. Now,computing the utility with respect to a single SCC assignment to a userinvolves determining the allocation of RBs to users based on the currentCC allocations and is still a hard problem due to the finite bufferconstraints of users. This leads us to the third level of decisionmaking (common to all users), whereby given an assignment of CCs tousers and their finite buffers, we iteratively pick an RB in a CC alongwith a user allocation that yields the highest marginal utility (steps15-21).

Thus, back-tracking the three levels of decision-making, the algorithmdetermines the assignment of CCs to a user in each iteration. Note thatalthough PCC is fixed for LTE users, the order in which LTE users arepicked by the scheduler impacts the assignment of SCCs to LTE-A users.Once the CC assignment to all users is complete, the RB allocation ineach CC can be obtained by computing the utility corresponding to thefinal assignment (step 13). While the for loops (steps 4-12) run inO(K²Nn), the core component of utility computation in each iteration(steps 15-21) runs in O(N²M²K), resulting in a net time complexity ofO(K³N³M²n).

In establishing a performance guarantee for GCA-FD, we will invokenested sub-modularity, and leverage the following result from [1], [2].

Lemma 4.1 If the incremental oracle is only a—approximable, then theapproximation guarantee of greedy sub-modular maximization changes to

$\frac{\alpha}{p + \alpha},$where the maximization is subject to a p—independence system.We now have the following result.Theorem 4.2 GCA-FD provides an approximation guarantee of ¼.

Proof. The proof is based on nested sub-modularity, where we will showthat the three levels of decision making in GCA-FD correspond to threesub-modular maximization (SM) problems, each nested within the other.

Level 1: Specifically, at the highest level, we have the following SMproblem.Ψ={(u,

c _(u,1) , . . . ,c _(u,n−1)

):∀uε

_(lteA) ,∀c _(u,j)ε[1,N]\P _(u)}∪{(u,P _(u)):∀uε

_(lte)}φ_(u)={(u,

c _(u,1) , . . . ,c _(u,n−1)

):∀c _(u,j)ε[1,N]\P _(u)}, if uε

_(lteA)={(u,P _(u))}, if uε

_(lte)

Now, the set of all feasible schedules S (tuples of user and CCassignments) would correspond to a partition matroid of Ψ, whereby therecan be at most only one element from each φ_(u) for any AεS (for LTEusers the element is fixed). Note that, while PCC is given for allusers, it is still considered as an element for LTE users to includethem in scheduling. Further, the scheduling objective function,

f(A) = ∑_(k ∈ ??)μ_(k)(A)corresponds to the aggregate user utility resulting from the given CCassignment in A as well as finite user buffers. Now, it can be seenthat,f(A∪(u,{right arrow over (c _(u))}))−f(A)≦f(B∪(u,{right arrow over (c_(u))}))−f(B), where, B⊂A

Since one user is considered at a time, when more elements (users, A\B)are added to A, some of the existing CCs will have more users in A thanin B. This in turn reduces the contribution of a new element to A morethan to B. In addition to being non-decreasing, incorporating all usersin the CC assignment process allows for normalizing f(f(Ø)=0)—althoughthe PCC assignment for LTE users is fixed, it impacts the SCC assignmentfor LTE-A users due to finite buffer scheduling. Thus, our schedulingobjective function is non-decreasing and sub-modular on S. Note thatconsidering all SCC assignments to a LTE-A user before moving to anotheruser is crucial for sub-modularity to hold with finitebuffers—otherwise, existing users whose allocations are replaced on someRBs by a new user, will have their freed up data available for laterallocations that could result in a higher marginal utility. Since GCA-FDemploys a greedy algorithm (step 10) to maximize this non-decreasing, SMobjective, its sub-optimality would be bounded by ½ [13], provided wecan optimally compute the incremental function (step 10),

$\begin{matrix}{\arg\;{\max\limits_{u \notin A}\left\{ {{f\left( {A\bigcup\left( {u,{\overset{\rightarrow}{c}}_{u}} \right)} \right)} - {f(A)}} \right\}}} & (3)\end{matrix}$Level 2: Computing the above incremental oracle is a hard problem initself, with an additional level of hardness arising from theexponential number of SCC assignments possible for an LTE-A user. GCA-FDapproximates it using the following SM problem for each LTE-A user.Ψ_(u)={(u ₁ ,c), . . . ,(u _(n 1) ,c):∀cε[1,N]}\

with, φ_(u) _(j) ={(u _(j) ,c):∀cε[1,N]}\

where the user u is replicated n−1 times (u_(j)=u, jε[1, n−1]). Now theset of all feasible schedules S_(u) (tuples of user and CC) for user uforms a partition matroid of Ψ_(u). With f(A_(u)) (A_(u)εS_(u))representing the same objective function as before, we can claim thatf(A _(u)∪(u _(j) ,c)−f(A _(u))≦f(B _(u)∪(u _(j) ,c))−f(B _(u))where B_(u) ⊂A_(u). This is because, when a new CC is available for useru (assume u's buffer is already used up without loss of generality), wecan move some of u's allocation on RBs in existing CCs to the new one,provided we have additional un-used buffer for other users in A forthese vacated RBs. Now by adding a CC later in the schedule, we run therisk of using up more of the remaining buffer of other users in A duringreplacement on other CCs that were added prior to it. Hence, u's abilityto replace RBs in the new CC and increase utility is reduced if addedlater in the schedule, resulting in f being sub-modular on S_(u).Further, f is non-decreasing and can be normalized to the utility priorto the consideration of user u. Now GCA-FD employs a greedy algorithm(step 7) to maximize this non-decreasing, SM objective. However,computing the incremental oracle arg max_(u) _(j,) _(c){f(A_(u)∪(u_(j),c))−f(A_(u))} in level 2 still remains a hard problem since it involvesdetermining the optimal allocation of RBs to users in CCs, subject tothe CC assignment in A_(u)∪(u_(j),c) as well as finite user buffers,which brings us to the final level.Level 3: Computing the incremental oracle in level 2 (level 1) for LTE-A(LTE) users is an extension (to multiple CC) of the RB allocationproblem considered in [16] for a single CC with finite user buffers.Hence, its hardness follows from the hardness of the problem in [16].GCA-FD approximates the problem using the following SM problem given theCC assignment and finite buffers of all users.Ψ′={(u,c,m):∀(u,c)εA∪

,mε[1,M],}with, φ_(c,m)={(u,c,m):uεA∪

}Now the set of all feasible schedules S′ (RB allocations) for a given CCassignment to users forms a partition matroid of Ψ′. The schedulingobjective for any A′εS′ is given as,

${f\left( A^{\prime} \right)} = {\sum\limits_{u \in {A^{\prime}\bigcup{??}}}{\mu_{u}\left( A^{\prime} \right)}}$${where},{{\mu_{u}\left( A^{\prime} \right)} = {v_{u}\min\left\{ {{\sum\limits_{{({c,m})}:{{({u,c,m})} \in A^{\prime}}}r_{u,m,c}},B_{u}} \right\}}}$

It is easy to see that f(A′∪(u,c,m))−f(A′)≦f(B′∪(u,c,m))−f(B′), where,B′⊂A′ since the assignment of an RB on a CC to a user later in theschedule will bring potentially lesser utility due to reduced bufferavailability (from prior RB allocations). Now, the greedy algorithm inGCA-FD that maximizes this non-decreasing, SM objective (step 18) woulddirectly yield an approximation of ½ since f(A′) can be computedoptimally.

Since the incremental oracle in level 2 can be solved with ½approximation (from level 3), using the result from lemma 4.1 andapplying p=1 for matroids, we obtain that level 2 itself can be solvedwith

$\frac{1/2}{1 + {1/2}} = \frac{1}{3}$approximation. However, since level 2 solves the incremental oracle forlevel 1, we obtain the final approximation for level 1 and hence thewhole algorithm as

$\frac{1/3}{1 + {1/3}} = {\frac{1}{4}.}$

V. Contiguous Carrier Aggregation

Constraining users with contiguous CC assignment helps energy efficiencybut reduces the flexibility in diversity scheduling. However, it alsoreduces the number of assignment possibilities for a user, therebysimplifying the assignment problem.

A. Backlogged Buffers

With the PCC already assigned to users, the assignment of SCC to LTE-Ausers may now be contiguous as well as include the PCC. Hence, there areat most only N−n+1 such contiguous CC combinations for every LTE-A user.We now propose the following algorithm (GCA-BC). The algorithm issimilar to the one for discontiguous CA with backlogged butlers (GCA-BD)with the following difference. Instead of assigning one SCC at a time,we now assign all SCCs jointly to a user, owing to the limited number ofcontiguous CC configurations that include the PCC (N−n+1 combinations,step 3). Hence, at each iteration, we select a user with his n CCassignment that yields the highest marginal utility (steps 5-7). Onceall users have received their CC assignments, we allocate the RBs in CCsto users yielding

Algorithm 3 Greedy Scheduler for SCA-BC: GCA-BC  1: Input: CC limit n;PCC List 

 = {(k, P_(k))}, ∀k; rate r_(k,m,c), ∀k, m, c  2:

 ←

_(lteA) ∪

_(lte);

_(lteA)← {1, . . . ,|

_(lteA) |},

_(lte) ← {|

_(lteA) | +1, . . . , |

 |};

 ← Ø  3: Define φ_(u) ← {(u, [c, c + n − 1]) : P_(u) ∈ [c, c + n − 1], c∈ [1, N − n + 1]}, ∀u ∈

_(lteA)  4: while

_(lteA) ≠ Ø do  5:  f(

) = Σ_(c)Σ_(m=1) ^(M) max_(u:(u,c)∈ )

{v_(u)r_(u,m,c)}  6:  f(

 ∪ (u′, [c_(u) ^(′), c_(u) ^(′) + n −1])) =  Σ_(c)Σ_(m=1) ^(M)max_(u:(u,[c) _(u) _(,c) _(u) _(+ n−1])∈)

_(∪(u′,[c) _(u) ^(′) _(,c) _(u) ^(′) _(+ n−1])){v_(u)r_(u,m,c)}  7: (u*, [c*, c* + n − 1]) = arg max_((u′,[c′,c′+n −1])∈φ) _(u′) _(:u′∈ )

_(lteA)  {f(

 ∪ (u′, [c′, c′ + n − 1])) − f(

)}  8:  

 ←

 ∪ (u*,[c*, c* + n − 1]);

_(lteA) ←

_(lteA)\u*  9: end while 10: k_(m,c)* = arg max_(k:(k,c)∈)

_(∪ )

{v_(k)r_(k,m,c)}, ∀m, cthe highest weighted rate on that RB and assigned to that CC (step 10).The bulk of the complexity comes from step 7, where the marginal utilitycalculation for a given new user assignment on n CCs incurs O(nM), withO(K(N−n+1)) such assignments possible, resulting in O(KNMn). Thiscoupled with O(K) iterations, results in a net complexity of O(K²NMn).The limited number of contiguous CC assignments results in a factor ncomplexity reduction compared to the discontiguous CA case.Theorem 5.1: GCA-BC provides an approximation guarantee of ½.

Proof: The proof is similar to that of GCA-BD, for a slightly differentdefinition of partition matroid.

Define the ground set as follows.Ψ={(u,[c,c+n−1]):P _(u) ε[c,c+n−1],∀cε[1,N−n+1],uε

_(lteA)}φ_(u)={(u,[c,c+n−1]):P _(u) ε[c,c+n−1],cε[1,N−n+1]}

Now, the set of all feasible schedules S (tuples of user and contiguousCC assignments) would correspond to a partition matroid of Ψ, wherebythere can be at most only one element from each φu for any AεS. Further,our scheduling objective function, f(A)=Σ_(cεC)μ_(c)(A), where

$\begin{matrix}{{\mu_{c}(A)} = {\sum\limits_{m}{\max\limits_{k:{{({k,c})} \in {{??}\bigcup{??}}}}\left\{ {v_{k}r_{k,m,c}} \right\}}}} & (4)\end{matrix}$can be shown to be non-decreasing and sub-modular on S using an argumentsimilar to that in the backlogged buffer case, and can be computedoptimally. Hence, the result.B. Finite Buffers

Our algorithm for contiguous CA with finite buffers (GCA-FC) is similarto its discontiguous counter-part, namely GCA-FD, with the followingdifference: the exponential number of discontiguous CC assignments toevery user now reduces to just N−n+1 contiguous CC assignments. Thisreduces the 3 levels of decision making in GCA-FD to 2 levels in GCA-FCfor all users. The first level remains to be the selection of the userwith its CC assignment (SCCs for LTE-A user, 1 fixed PCC for all users)that yields the highest marginal utility in each iteration (steps 4-7).However, the second level of determining the set of SCCs in anassignment to the LTE-A user (steps 5-9 in GCA-FD) is completelybypassed given the fixed number of contiguous CC assignments (step 3).This directly leads us to the third level, where for a given assignmentof CCs to users and their finite buffers, we iteratively pick an RB in aCC along with a user allocation that yields the highest marginal utility(similar to steps 15-21 in GCA-FD). The utility computation itself runsin O(N²M²K) as before, of which there are O(KN) user-CC assignmentspossible to select from. This along with K iterations, results in a nettime complexity of O(K³N³M²). Once again, restricting to contiguous CCassignments brings in a complexity reduction factor of n (compared toGCA-FD).

Algorithm 4 Greedy Scheduler for SCA-FD: GCA-FC 1: Input: CC limit n;PCC List 

 = {(k, P_(k))}, ∀k ; rate r_(k,m,c), ∀k, m, c; Buffer limit B_(k) , ∀k2:

 ←

_(lteA) ∪

_(lte);

_(lteA) ← {1, . . . , |

_(lteA) |},

_(lte) ← {|

_(lteA) | +1, . . . ,|

 |};

 ← Ø 3: Define φ_(u) ← {(u, {right arrow over (c)}_(u)) : ∀u ∈

; {right arrow over (c)}_(u) ^(′) = [c, c + n − 1] : P_(u) ∈ [c, c + n −1], c ∈ [1, N − n + 1]}, ∀u ∈

_(lteA); {right arrow over (c)}_(u) = P_(u), ∀u ∈

_(lte) 4: while

 ≠ Ø do 5:  (u, {right arrow over (c)}_(u))* = arg max_((u,c) _(u) ^(′)_()∈φ) _(u) _(:u∈ )

 {f(

 ∪ (u, {right arrow over (c)}_(u))) − f(

)} 6:  

 ←

 ∪ (u, c_(u) ^(′) )*;

 ←

\u* 7: end while 8: Compute f(

) to obtain RB allocation in each CC (similar to GCA-FD).Theorem 5.2: GCA-FC provides an approximation guarantee of ₃⅓.Proof: The proof is along the lines of that for GCA-FD. The ground sethowever, incorporates only contiguous CC assignments.Ψ={(u,[c,c+n−1]):∀uε

_(lteA) ,P _(u) ε[c,c+n−1],∀cε[1,N−n+1]}∪{(u,P _(u)):∀uε

_(lte)}φ_(u)−{(u,[c,c+n−1]):P _(u) ε[c,c+n−1],∀cε[1,N−n+1]}, if uε

_(lteA)={(u,P _(u))}, if uε

_(lte)

The definitions of Ψ and φ_(u) are essentially subsets of those in thediscontiguous case, where all possible SCC combinations were allowed.Hence, the rest of the arguments for sub-modularity hold good given thatthe objective function is the same. However, unlike GCA-FD, we have onlyone level of nested sub-modularity. The computation of the incrementaloracle (in step 5) amounts to the determination of RB allocations tousers given their CC assignment and finite buffers. Latter being anon-decreasing SM problem, results in a half approximation, which inturn results in a net approximation of ½/(1+½)=⅓ (using lemma 4.1) forthe complete algorithm.

VI. Performance Evaluation

A. Set-Up

System Parameters: A frame-level simulator written in C++ is consideredfor evaluation of the proposed algorithms. A single-cell OFDMA downlinksystem based on LTE-A is considered, with a cell radius of 600 m. BSoperates at P_(BS)=35 dBm power. MS are uniformly distributed within thecell. The Okumura-Hata urban path loss model is employed and coupledwith log-normal shadowing and fast (Rayleigh) fading. Each user'sRayleigh channel has a Doppler fading equivalent to a velocity of 3-10Km/hour. We consider constant bit rate (CBR) applications as thegenerators of traffic. We consider an LTE-A downlink frame of 1 ms,which consists of two slots (sub-frames) of 0.5 ms each. Each slotconsists of 7 symbols. Resource allocations to users are made at thegranularity of physical resource blocks (PRB). Each PRB consists of 12sub-carriers of 15 KHz each. The number of PRBs in the system scaleswith the available bandwidth (25 for 5 MHz, 50 for 10 MHz, etc.). Of the14 symbols per frame, 11 are available for data. Given a maximum of fivebits (64-QAM) that can be loaded in each symbol, the peak downlink(SISO) data rate for a 5 MHz CC is 16.5 Mbps, which in turn can getboosted to 82.5 Mbps by aggregating 5 such CCs. The rate supported by auser on a RB is obtained by mapping the received SNR at the user to oneof the 27 MCS values in LTE-A.

Baselines and Metrics: We compare our proposed algorithms for joint CCselection and scheduling with two baselines: Load-based and RSRP-based,wherein the SCC selection is made based on metrics outlined in SectionIII-A for PCC selection, while scheduling itself (given a CC assignment)follows the same approach as in our proposed algorithms. The schedulingalgorithms are evaluated per frame, where the main metric of evaluationis aggregate network throughput (Vk=1, ∀k). The results are averagedover twenty topologies. The limit on the # SCCs, number of users,fraction of LTE users, and user buffers are the parameters ofevaluation. The default parameters of operation (when not specified)include a system with: 10 users (all LTE-A users) with backloggedbuffers and 5 total CCs (each with 5 Mhz spectrum in the 2.1 GHz band)with a limit of 2 CCs per LTE-A user.

B. Performance of Joint CC Selection and Scheduling

While limiting the number of SCCs helps reduce energy consumption forLTE-A users, we need to understand if the joint CC selection andscheduling will keep the degradation in throughput performance small. Weconsider the backlogged buffer case, where the problem is optimallysolvable when there is no limit on the # CCs (assign all CCs to everyLTE-A user). We compare the loss in throughput of our integrated (GRA-BDand GRA-BC) algorithms for discontiguous (dis) and contiguous (con) CArespectively due to limited # CCs against the optimal (allowing for allCCs). The result in FIG. 2(a) clearly shows that our algorithms helpachieve 80-90% of the maximum performance in practice (compared to theworst case guarantees of half) while operating on just half the numberof CCs. Thus, with the help of efficiently designed algorithms for jointCC selection and scheduling, energy reduction for LTE-A users can berealized without sacrificing any appreciable throughput performance.

We also evaluate the benefit of integrating CC selection with schedulingby comparing our algorithms against decoupled CC selection andscheduling schemes (Load-based and RSRP-based CC selection) in FIGS.2(b) and (c). Topologies with five and ten CCs are considered with fiveand ten LTE-A users respectively. Five observations can be made: (i) Inthe region of practical interest (limiting CCs to 2-3), we see thatgains of 15-30% can be had from integrating CC selection withscheduling; (ii) gains for contiguous CA are slightly inferior to thoseof discontiguous CA as contiguous CC selection restricts schedulingdiversity gain; (iii) gains decrease as the limit on # CCs is relaxedsince the need for careful CC selection diminishes in this case; (iv)gains are more over the RSRP-based scheme, indicating thatload-balancing users across CCs is relatively more beneficial when CCselection is decoupled from scheduling; and (v) more CCs (10) providesincreased room for scheduling diversity and hence higher gains.

In contrast to FIG. 2(a), we now fix the limit on the # CCs and studythe loss in performance as a function of the # users in FIG. 2(d). Itcan be seen that to keep the loss in performance small, the system needssufficient scheduling diversity. This can be achieved either with alarger limit on # CCs, or with a larger number of users. Hence, for atotal of 5 CCs, a smaller limit (2) on CCs requires more users (10) toachieve over 90% of optimal performance, while a limit of 3 CCs requiresonly 5 users. Further, contiguous CA reduces the ability to leveragescheduling diversity. This coupled with a larger # of CCs (10), requireseither a larger limit (3) on CCs or a larger number of users (20) toachieve over 90% of optimal performance.

C. Impact of User Heterogeneity and Finite Buffers

We study the impact of user heterogeneity (coexistence of LTE-A and LTEusers) on the performance of CA as an increasing function of LTE users.The algorithms employed are GCA-BD and GCA-BC for the backlogged buffercase. FIG. 3(a) indicates that when the system transitions from beingpredominantly LTE-A to predominantly LTE, the loss in performance can beas high as 30%. Access to more CCs allows LTE-A users to provide morescheduling diversity. Note that the loss in performance in FIG. 3(a) isonly due to diminishing gain in scheduling diversity across CCs sinceour fairness model (in scheduling) computes weights only based on netthroughput of LTE-A users and not their per-CC throughput. If fairnessbased on per-CC throughput is employed, then one can expect the loss inperformance with increasing number of LTE users to further increase.

The gain of our algorithms over the load-based scheme is studied as afraction of the LTE users in the system in FIG. 3(b). Two observationscan be made: (i) Irrespective of the fraction of LTE users, gains of15-20% can be had. This is because our algorithms employ a PCC selectionmechanism that incorporates both load and RSRP metrics, along with ajoint SCC selection and scheduling scheme. While the impact of thelatter component reduces with increasing # of LTE users, the impact ofthe former component increases as it applies to LTE users; (ii) The gainsurprisingly increases with increasing fraction of LTE users forcontiguous CA to as high as 25-35%. Recall that the ability to leveragescheduling diversity is lesser with contiguous CA. Hence, when thenumber of LTE-A users is very small, the need to carefully select theirSCCs is all the more pronounced.

We finally study the impact of finite buffers on our algorithms, GCA-FDand GCA-FC and compare them against the baselines in FIGS. 3(c) and (d).When the amount of user buffer is small, it becomes the performancebottleneck, leaving little room for any performance optimization andgains. However, as the buffer size increases, there is more room toleverage the increased scheduling diversity gain with CA, resulting inhigher gains. The rest of the inferences comparing our algorithms forfinite buffer against the baselines are similar to those in thebacklogged buffer case.

Referring to FIG. 4, the system picks the primary CC for a new user as

$c_{p} = {\arg\;{\max_{c}\frac{r_{{avg},c}}{l_{c}}}}$in step 1 (block 402). Here, (r_(k,c) ^(avg) ¹ is the average referencesignal received power (RSRP) averaged over the spectrum for user ‘k’ oncomponent carrier ‘c’. l_(c) is the load on carrier c and couldrepresent the number of users already assigned to carrier c or thetraffic demand on carrier c, etc. Then, the system proceeds with step 2if joint CC selection and scheduling do not apply (block 404), andselects secondary CCs up to the allocated limit for a new LTE advanceduser based on the same criteria as for PCC in step 2 (block 406). Thesystem proceed with step 3 in case of joint CC selection and scheduling(block 404), and performs joint secondary CC selection (for LTE advancedusers) and resource scheduling (for all users) in step 3 (block 408).The process of joint CC selection and scheduling in step 3 (block 408)would vary depending on whether the users have finite user buffers orbacklogged user buffers. We describe the approach for both these casesseparately.

An approach for the backlogged user buffers is shown in FIG. 5. In step1 (block 502), the system picks a user and CC pair such that assigningthe CC as a secondary to the LTE-adv user yields the highest incrementalvalue to the current schedule. In step 1 (block 502), the systemdetermines the incremental value of a CC assignment to a user amounts tofinding the best utility/metric (eg. weighted rate) on each resourceblock (RB) in a CC based on the users who are assigned to that CC andaggregating them. Further in step 1 (block 502), if the CCs that areassigned to a LTE-adv user are constrained to be contiguous, instead ofassigning one CC at a time, the solution would consider all combinations(N−n+1) of ‘n’ CCs to an LTE-adv user in determining the best set of n−1CCs that need to be assigned to the LTE-adv user. Here ‘n’ is the limiton number of secondary CCs that can be assigned to an LTE-adv user outof a total of N. Then, in step 2 (block 504), the system continues theassignment till all LTE-adv users have their limited share of secondaryCCs assigned.

An approach for the finite user buffers is shown in FIG. 6. In step 1(block 602), the system picks a user along with its set of n−1 secondaryCC assignments such that assigning the set of n−1 secondary CCs to theLTE-adv user yields the highest incremental value to the currentschedule. In step 1a (block 604), the system picks secondary CCs one byone iteratively for the given user such that addition of the new CCprovides the highest incremental value. In step 1ai (block 606), thesystem determines the value of assigning a CC to a user by allocatingRBs in the CC to users based on current CC assignments: at each step theRB yielding the highest incremental value to a user taking its finitebuffer into account is selected and assigned to the user. Then, in step2 (block 608), the system continues the assignment till all LTE-advusers have their set of secondary CCs assigned.

In this document, we have looked at the problem of energy efficientcarrier aggregation for LTE-A systems. We have shown that with the helpof efficiently designed joint CC selection and scheduling algorithms,one can obtain close-to the maximum scheduling performance of CAsystems, while expending only half the energy for its mobile devices. Inthis regard, we proposed algorithms with performance guarantees undervarious models of contiguous and discontiguous CA as well as backloggedand finite user buffers. Our extensive evaluations validated ourhypothesis as well as highlighted the performance gains of our proposedsolutions over baseline schemes.

The foregoing is to be understood as being in every respect illustrativeand exemplary, but not restrictive, and the scope of the inventiondisclosed herein is not to be determined from the Detailed Description,but rather from the claims as interpreted according to the full breadthpermitted by the patent laws. It is to be understood that theembodiments shown and described herein are only illustrative of theprinciples of the present invention and that those skilled in the artmay implement various modifications without departing from the scope andspirit of the invention. Those skilled in the art could implementvarious other feature combinations without departing from the scope andspirit of the invention.

What is claimed is:
 1. A method implemented in a mobile communicationssystem, comprising: selecting primary component carrier c_(p) for a newuser according to formula,$c_{p} = {\arg\;{\max_{c}\frac{r_{k,c}^{avg}}{l_{c}}}}$ where r_(k,c)^(avg) is reference signal received power (RSRP) averaged over aspectrum for user k on component carrier c; performing joint secondaryCC selection for one or more Long Term Evolution (LTE)-Advanced usersand resource scheduling for one or more LTE users and said one or moreLTE-Advanced users; wherein the performing includes: selecting anLTE-Advanced user according to a set of n−1 secondary CC assignments sothat assigning a set of n−1 secondary CCs to the LTE-Advanced usermaximizes an incremental value to an existing schedule; selecting eachof one or more secondary CCs iteratively for a given user so that addingsaid each of one or more secondary CCs maximizes an incremental value toan existing schedule; determining the incremental value by allocating aresource block (RB) in the secondary CC to the given user based on anexisting CC assignment; and selecting the RB so that the incrementalvalue is maximized.
 2. The method as in claim 1, wherein the performingcomprises: selecting a pair of an LTE-Advanced user and a secondary CCso that assigning the secondary CC to the LTE-Advanced user maximizes anincremental value to an existing schedule.
 3. The method as in claim 2,wherein the performing further comprises: continuing the assignmentuntil all of said one or more LTE-Advanced users have a share ofsecondary CCs.
 4. The method as in claim 1, wherein the selecting the RBis performed by taking a finite user buffer into account.
 5. The methodas in claim 1, further comprising: continuing the assignment until allof said one or more LTE-Advanced users have a set of secondary CCs. 6.The method as in claim 1, wherein the mobile communications system is abase station.
 7. A mobile communications system comprising: a firstselection unit to select primary component carrier c_(p) for a new useraccording to formula,$c_{p} = {\arg\;{\max_{c}\frac{r_{k,c}^{avg}}{l_{c}}}}$ where r_(k,c)^(avg) is reference signal received power (RSRP) averaged over aspectrum for user k on component carrier c; and a performance unit toperform joint secondary CC selection for one or more Long Term Evolution(LTE)-Advanced users and resource scheduling for one or more LTE usersand said one or more LTE-Advanced users; wherein the performance unitselects an LTE-Advanced user according to a set of n−1 secondary CCassignments so that assigning a set of n−1 secondary CCs to theLTE-Advanced user maximizes an incremental value to an existingschedule; wherein the performance unit further selects each of one ormore secondary CCs iteratively for a given user so that adding said eachof one or more secondary CCs maximizes an incremental value to anexisting schedule; wherein the performance unit further determines theincremental value by allocating a resource block (RB) in the secondaryCC to the given user based on an existing CC assignment, and selects theRB so that the incremental value is maximized.
 8. The mobilecommunications system as in claim 7, wherein the performance unitselects a pair of an LTE-Advanced user and a secondary CC so thatassigning the secondary CC to the LTE-Advanced user maximizes anincremental value to an existing schedule.
 9. The mobile communicationssystem as in claim 8, wherein the performance unit further continues theassignment until all of said one or more LTE-Advanced users have a shareof secondary CCs.
 10. The mobile communications system as in claim 7,wherein the selecting the RB is performed by taking a finite user bufferinto account.
 11. The mobile communications system as in claim 7,further performance unit further continues the assignment until all ofsaid one or more LTE-Advanced users have a set of secondary CCs.
 12. Themobile communications system as in claim 7, wherein the mobilecommunications system is a base station.